Abstract. In this work, we introduce and study a new model for selfish routing over non-cooperative networks that combines features from the two such best studied models, namely the KP model and the Wardrop model in an interesting way. We consider a set of n users, each using a mixed strategy to ship its unsplittable traffic over a network consisting of m parallel links. In a Nash equilibrium, no user can increase its Individual Cost by unilaterally deviating from its strategy. To evaluate the performance of such Nash equilibria, we introduce Quadratic Social Cost as a certain sum of Individual Costs -namely, the sum of the expectations of the squares of the incurred link latencies. This definition is unlike the KP model, where Maximum Social Cost has been defined as the maximum of Individual Costs. We analyse the impact of our modeling assumptions on the computation of Quadratic Social Cost, on the structure of worst-case Nash equilibria, and on bounds on the Quadratic Coordination Ratio.
We consider the problem of routing n users on m parallel links under the restriction that each user may only be routed on a link from a certain set of allowed links for the user. So, this problem is equivalent to the correspondingly restricted scheduling problem of assigning n jobs to m parallel machines. In a Nash equilibrium, no user may improve its own Individual Cost (latency) by unilaterally switching to another link from its set of allowed links.For identical links, we present, as our main result, a polynomial time algorithm to compute from any given assignment a Nash equilibrium with non-increased makespan. The algorithm gradually transforms the assignment by pushing the unsplittable user traffics through a flow network, which is constructed from the users and the links. The algorithm uses ideas from blocking flows.Furthermore, we use techniques simular to those in the generic PREFLOWPUSH algorithm to approximate in polynomial time a schedule with optimum makespan. This results to an improved approximation factor of 2 − 1 w1 for identical links, where w 1 is the largest user traffic, and to an approximation factor of 2 for related links.
A Nash equilibrium of a routing network represents a stable state of the network where no user finds it beneficial to unilaterally deviate from its routing strategy. In this work, we investigate the structure of such equilibria within the context of a certain game that models selfish routing for a set of n users each shipping its traffic over a network consisting of m parallel links. In particular, we are interested in identifying the worst-case Nash equilibrium-the one that maximizes social cost. Worst-case Nash equilibria were first introduced and studied in the pioneering work of Koutsoupias and Papadimitriou [9]. More specifically, we continue the study of the Conjecture of the Fully Mixed Nash Equilibrium, henceforth abbreviated as FMNE Conjecture, which asserts that the fully mixed Nash equilibrium, when existing, is the worst-case Nash equilibrium. (In the fully mixed Nash equilibrium, the mixed strategy of each user assigns (strictly) positive probability to every link.) We report substantial progress towards identifying the validity, methodologies to establish, and limitations of, the FMNE Conjecture.
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